Masaaki Homma

Kanagawa University · Department of Mathematics and Physics

We determine the minimal bi-degree(s) of an irreducible filling curve over Fq for P1×P1 by constructing examples. It is (q+1,q+1) if q≠2, and they are (4, 3) and (3, 4) if q = 2.

We determine the minimal bi-degree(s) of an irreducible filling curve over $\mathbb{F}_q$ for $\mathbb{P}^1\times \mathbb{P}^1$. It is $(q+1, q+1)$ if $q\neq 2$, and they are $(4,3)$ and $(3,4)$ if $q=2$.

The classification of maximal plane curves of degree $3$ over $\mathbb{F}_4$ will be given, which complements Hirschfeld-Storme-Thas-Voloch's theorem on a characterization of Hermitian curves in $\mathbb{P}^2$. This complementary part should be understood as the classification of Sziklai's example of maximal plane curves of degree $q-1$ over $\math...

In this article we prove a conjecture formulated by A. B. Sørensen in 1991 on the maximal number of F q 2 \mathbb {F}_{q^2} -rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree d ≤ q . d \le q.

In this paper we characterize the non-singular Hermitian variety ${\mathcal H}(6,q^2)$ of $\mathrm{PG}(6, q^2)$, $q\neq2$ among the irreducible hypersurfaces of degree $q+1$ in $\mathrm{PG}(6, q^2)$ not containing solids by the number of its points and the existence of a solid $S$ meeting it in $q^4+q^2+1$ points.

In this article we prove a conjecture formulated by A.B. Soerensen in 1991 on the maximal number of $\mathbb{F}_{q^2}$-rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree $d \le q.

Nonsingular plane curves over a finite field Fq of degree q+2 passing through all the Fq-points of the plane admit a representation by 3×3 matrices over Fq. We classify their degenerations by means of the matrix representation. We also discuss the similar problem for the affine-plane filling projective curves of degree q+1.

Nonsingular plane curves over a finite field $\mathbb{F}_q$ of degree $q+2$ passing through all the $\mathbb{F}_q$-points of the plane admita representation by $3\times 3$ matrices over $\mathbb{F}_q$. We classify their degenerations by means of the matrix representation, and also discuss the similar problem for the affine-plane filling curves of d...

In Homma and Kim (2010), an upper bound of the number of rational points on a plane curve of degree d d over F q F q is found. Some examples attaining the bound are given in Homma and Kim (2010), whose degrees are q+2 q + 2 , q+1 q + 1 , q q , q−1 q − 1 , q − 1 (when q q is a square), and 2 2 . In this paper, we consider an actual upper bound on su...

The numbers of $\mathbb{F}_q$-points of nonsingular hypersurfaces of a fixed degree in an odd-dimensional projective space are investigated, and an upper bound for them is given. Also we give the complete list of nonsingular hypersurfaces each of which realizes the upper bound. This is a natural generalization of our previous study of surfaces in p...

In 1990, Hefez and Voloch proved that the number of $F_q$-rational points on a nonsingular plane $q$-Frobenius nonclassical curve of degree $d$ is $N = d(q-d+2)$. We address these curves in the singular setting. In particular, we prove that $d(q-d + 2)$ is a lower bound on the number of $F_q$-rational points on such curves of degree $d$.

A basis of the ideal of the complement of a linear subspace in a projective space over a finite field is given. As an application, the second largest number of points of plane curves of degree $d$ over the finite field of $q$ elements is also given for $d\geq q+1$.

We give an upper bound for the number of points of a hypersurface over a finite field that has no lines on, in terms of the dimension, the degree, and the number of the elements of the finite field.

A nonsingular surface of degree $d \geq 2$ in $\mathbb{P}^3$ over $\mathbb{F}_q$ has at most $((d-1)q+1)d$ $\mathbb{F}_q$-lines, and this bound is optimal for $d = 2, \sqrt{q}+1, q+1$.

In the previous paper, we established an elementary bound for numbers of points of surfaces in the projective $3$-space over ${\Bbb F}_q$. In this paper, we give the complete list of surfaces that attain the elementary bound. Precisely those surfaces are the hyperbolic surface, the nonsingular Hermitian surface, and the surface of minimum degree co...

We establish an upper bound for the number of points of a hypersurface without a linear component over a finite field, which is analogous to the Sziklai bound for a plane curve.Our bound is the best one for irreducible hypersurfaces that is linear on their degrees, because, for each finite field, there are at least two irreducible hypersurfaces of...

The nonsingular Hermitian surface of degree $\sqrt{q} +1$ is characterized by its number of $\Bbb{F}_q$-points among the irreducible surfaces over $\Bbb{F}_q$ of degree $\sqrt{q} +1$ in the projective 3-space.

We prove the uniqueness of a plane curve of degree q over a finite field FqFq which attains Sziklaiʼs bound q(q−1)+1q(q−1)+1. More precisely, if a plane curve of degree q over FqFq has q(q−1)+1q(q−1)+1 rational points, then it is projectively equivalent to the curve defined by the equation Xq−XZq−1+Xq−1Y−Yq=0Xq−XZq−1+Xq−1Y−Yq=0. Although the case q...

For a nondegenerate irreducible curve $C$ of degree $d$ in ${\Bbb P}^r$ over ${\Bbb F}_q$ with $r \geq 3$, we prove that the number $N_q(C)$ of ${\Bbb F}_q$-points of $C$ satisfies the inequality $N_q(C) \leq (d-1)q +1$, which is known as Sziklai's bound if $r=2$.

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a generalization of the curve are also presented.

For a plane curve over FqFq of degree q+1q+1, it is known by our previous work that the number of its FqFq-rational points is at most q2+1q2+1. In this paper, we determine the curves that attain this maximum, up to projective equivalence.

We manage an upper bound for the number of rational points of a Frobenius nonclassical plane curve over a finite field. Together with previous results, the modified Sziklai conjecture is settled affirmatively.

This paper has double purposes. One of them is to give a new bound on the number of points of a plane curve of degree d over a finite field that meets Sziklai's conjectural bound at d=q+1. An example shows that this bound is sharp for d=q+1. The second one is to study an example against that conjecture for q=d=4. This curve also shows the sharpness...

We settle the conjecture posed by Sziklai on the number of points of a plane curve over a finite field under the assumption that the curve is nonsingular. Comment: 12 pages

The aim of this article is the determination of the second generalized Hamming weight of any two-point code on a Hermitian curve of degree q + 1. The determination involves results of Coppens on base-point-free pencils on a plane curve. To avoid non- essential trouble, we assume that q > 4.

Our concern is a nonsingular plane curve defined over a finite field of q elements which includes all the rational points of the projective plane over the field. The possible degree of such a curve is at least q+2. We prove that nonsingular plane curves of degree q+2 having the property actually exist. More precisely, we write down explicitly all o...

For a Hermitian curve H in projective plane P 2 and an arbitrary point P of P 2 , we find out the Galois group of the projection H → P 1 with center P . To achieve this aim, we discuss the Galois group of an equation and that of a finite separable morphism between curves in slightly more general context. Moreover, we compute the genus of the so-cal...

This is the second part of the series of papers devoted to the determination of the minimum distance of two-point codes on a Hermitian curve. We study the case where the minimum distance agrees with the designed one. In order to construct a function which gives a codeword with the designed minimum distance, we use functions arising from conics in t...

The description of Galois points with respect to a Hermitian curve is given, which suggests that Yoshihara's theory of Galois points needs modifying if the characteristic of the ground field is positive.

We find all candidates for the Weierstrass semigroup of a pair of points on a cyclic 4-gonal curve which are total ramification points of the fixed cyclic 4-gonal map. Then we prove that such semigroups actually occur on some cyclic 4-gonal curve by constructing cyclic 4-gonal curves explicitly. Moreover, we characterize all Weierstrass semigroups...

This is a continuation of the previous papers [3, 4, 5]. We finish determining the minimum distance of two-point codes on a Hermitian curve.

In Homma M and Kim SJ [2], the authors considered two-point codes on a Hermitian curve defined over fields of odd characteristic. In this paper, we study the geometry of a Hermitian curve over fields of even characteristic and classify the two-point codes whose minimum distances agree with the designed ones.

This is a first step toward the determination of the parameters of two-point codes on a Hermitian curve. We describe the dimension of such codes and determine the minimum distance of some two-point codes.

In our previous paper (Bull. Korean Math. Soc. 37(2000), 493-505), we claimed a theorem on a certain subset of a projective space over a finite field (Theorem 3.1). Recently, however, Professor Kato pointed out that our proof does not work if the field consists of two elements. Here we give an alternative proof of the theorem for the exceptional ca...

The order of speciality of an ample invertible sheaf L on a curve is the least integer m so that L ⊗m is nonspecial. There is a reasonable upper bound of the order of speciality for a simple invertible sheaf in terms of its degree and projective dimension. We study the case where it reaches the upper bound. Moreover we for-mulate Castelnuovo's genu...

We started on a systematic investigation of the Weierstrass pairs on a smooth curve, in our previous papers (Arch. Math. 62 (1994) 73–82); 67 (1996) 337–348. We push our study further for the purpose of constructing Goppa codes with good parameters.

We prove that for a finite covering of curves the Clifford index of the source is at least that of the target.

We discuss the class of projective systems whose sup-ports are the complement of the union of two linear subspaces in general position. We express the weight enumerators of the codes generated by these projective systems using two simplex codes cor-responding to given linear subspaces. We also prove these codes are uniquely determined upto equivale...

Using two divisors F and D on an algebraic curve, Goppa [G] introduced a new way to construct a linear code, and he estimated the main parameters of the code. Choosing more restricted divisors F and D than ones in the general construction of a Goppa code, we may expect to improve the lower bound of the minimum distance. Actually, Garcia and Lax [GL...

Singular curves with a morphism of degree two onto a projective line should be classified into two types according as the equipped morphism is separable or not; we call a curve with a separable one a hyperelliptic curve of separable type, and the other a hyperelliptic curve of inseparable type. We give concrete expressions of a hyperelliptic curve...

We prove two variants of a base-point-free pencil trick, which are similar in the spirit of the proof, and apply them to the study of special divisors on a smooth plane curve involving a theorem of Max Noether.

We study a particular geometry in characteristic 2. Let X be a cubic surface obtained from ℙ2 by means of blowing up with centersP 1,...,P 6. We prove that X is projectively equivalent to the Fermat cubic surface with equation ∑x i3 if and only if each point of P 1,..., P 6 is the nucleus of the conic passing through the remaining five points.

We are interested in a particular geometry of plane curves in characteristicp>0, which was inspired by Thas's article [13]. We will prove that any plane curve of degree > 2 whose tangent lines at collinear points are concurrent is either a strange curve or projectively equivalent to the Fermat curve of degreeq + 1, whereq is a power ofp.

We prove that for given two distinct smooth curves in the projective plane over an algebraically closed field, their duals in the dual plane coincide if and only if they are conies in characteristic two with the same center.

We prove that for given two distinct smooth curves in the projective plane over an algebraically closed field, their duals in the dual plane coincide if and only if they are conics in characteristic two with the same center.

We study order-sequences of linear systems on smooth curves and establish the formula:b j +b N−j ≤b N for allj, where {b 0<b 1<...<b N } is the order-sequence of a linear system on a curve. As an application of the formula, we describe all linear systems on curves which have no Weierstrass points.

We work mainly over a field of characteristic > 0. Let X be a curve in projective N- space. We study the connection between the gap sequence (associated with the embedding of X) at a general point of X and reflexivity of osculating developables of X.

An algebraic distance graph is defined to be a graph with vertices inEn in which two vertices are adjacent if and only if the distance between them is an algebraic number. It is proved that an algebraic distance graph with finite vertex set is completeif and only if the graph is "rigid". Applying this result, we prove that (1) if all the sides of a...

An algebraic distance graph is defined to be a graph with vertices in En in which two vertices are adjacent if and only if the distance between them is an algebraic number. It is proved that an algebraic distance graph with finite vertex set is complete if and only if the graph is "rigid". Applying this result, we prove that (1) if all the sides of...

http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=450530&tocid=100062582&page=189-198

Let фL: C[?]P^no(l)-1 be the projective embedding of a complete non-singular curve C of geneus g by means of Γ(L), where L is a very ample invertible sheaf on C. We will study the homogeneous coordinate ring and ... http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=450523&tocid=100040735&page=269-279

If an order q of an automorphism of an algebraic curve of genus g2 is prime, then q2g+1. In this paper we determine all curves having an automorphism of order 2g+1 when 2g+1 is prime.

Let C be a complete reduced irreducible curve of arithmetic genus g over an algebraically closed field K. Let L be a very ample invertible sheaf of degree d on C, and let фL: ... http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=450521&tocid=100040652&page=31-39

Let Γ be a congruence subgroup of SL(2, R), and let R(Γ)　be the Rimeann surface associated with Γ, i.e., R(Γ) is the canonical cimpatification of Γ\H, where H is the upper half plane of C. ... http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=450520&tocid=100040630&page=65-74

A Hermitian curve X is a plane curve of degree q + 1 which is projectively equiv- alent to the plane curve with the inhomogeneous equation yq + y = xq+1 over the finite field Fq2 of q2 elements, which has q3 +1 Fq2-rational points. The geometry of lines over Fq2 harmonizes with those points, that is to say, a line over Fq2 either tangents to X at a...

An invariant toward a classification of curves X with a birationally very ample and special invertible sheaf L is proposed, and an extremal pair (X,L) by means of the invariant is a smooth plane curve X with O X (1) corresponding line sections and vice versa. For pairs (X,L) next to extremal ones, all but one exception is characterized by the image...

Thesis--University of Tsukuba, D.Sc.(A), no. 92, 1981. 3. 25 https://www.tulips.tsukuba.ac.jp/limedio/dlam/B57/B576014/1.pdf